Stats Homework Questions

Question

1. In a mid-size company, the distribution of the number of phone calls answered each day by each of the 12 receptionists is bell-shaped and has a mean of 57 and a standard deviation of 11. Using the empirical rule, what is the approximate percentage of daily phone calls numbering between 24 and 90?

2. In a mid-size company, the distribution of the number of phone calls answered each day by each of the 12 receptionists is bell-shaped and has a mean of 50 and a standard deviation of 3. Using the empirical rule (as presented in the book), what is the approximate percentage of daily phone calls numbering between 44 and 56?

3.The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 62 ounces and a standard deviation of 6 ounces.

Use the Empirical Rule.

Suggestion: sketch the distribution in order to answer these questions.

a) 68% of the widget weights lie between _______ and _________

b) What percentage of the widget weights lie between 44 and 68 ounces?  %

c) What percentage of the widget weights lie below 74 ?  %

Answer 1

1. The mean is 57. The interval 24 to 90 is 57 +/- 33. Since the standard deviation is 11, then 57 +/- 33 is the same as 57 +/- 3 standard deviations. Per the empirical rule, 99.7% of the data is contained within 3 standard deviations of the mean.
2. The mean is 50. The interval 44 to 56 is 50 +/- 6. Since the standard deviation is 3, then 50 +/- 6 is the same as 50 +/- 2 standard deviations. Per the empirical rule, 95% of the data is contained within 2 standard deviations of the mean.
3. a. mean is 62, standard deviation is 6. Per the empirical rule, 68% of the data is contained within 1 standard deviation of the mean, so 68% of the data is between 62 +/ - 6, which is between 56 and 68.

b. 68-62 = 6, so 68 is one standard deviation above the mean and 62-44 = 18, so 44 is 3 standard deviations below the mean. Per the empirical rule, the percentage of data between 3 standard deviations below the mean to one standard deviation above the mean = 49% (mean to data 3 standard deviations below the mean) + 34% (mean to one standard deviation above the mean) = 83% of the data is between 44 and 68.

c. 74 - 62 = 12, so 74 is two standard deviations above the mean. Since the normal curve is symmetric about the mean, 50% of the data is below the mean of 62. Per the empirical rule, 47.5% of the data is within 2 standard deviations above the mean. Adding the percentages together, 50 + 47.5 = 97.5% of the data is below 74.